It is well known in the art that a first order temperature independent bandgap reference voltage VBGAP can be described as:
                    VBGAP        =                              VBE            ⁢                                                  ⁢            1                    +                      KD            ×            Vt            ⁢                                                  ⁢            ln            ⁢                                                  ⁢                          (                                                IC                  ⁢                                                                          ⁢                  1                  ×                  IS                  ⁢                                                                          ⁢                  2                                                  IS                  ⁢                                                                          ⁢                  1                  ×                  IC                  ⁢                                                                          ⁢                  2                                            )                                                          (        1        )            where: VBE1 is the base emitter voltage of a first bipolar transistor; IC1 is the collector current of the first bipolar transistor; IS1 is the saturation current of the first bipolar transistor; IC2 is the collector current of a second bipolar transistor; and IS2 the saturation current of the second bipolar transistor. Furthermore, IC1 is greater than IC2 because the current or the current density of the first bipolar transistor has to be greater than that of the second bipolar transistor. VT is the temperature voltage; KD is a design parameter greater than 1; and n is the ratio between IC1 and IC2. IS1 and IS2 are usually equal. Equation (1) can also be written as:VBGAP=VBE1+KD(VBE1−VBE2)  (2)where: VBE2 is the base emitter voltage of the second bipolar transistor. The bandgap voltage VBGAP is a technology dependent constant and about 1.2 V.
Equation (2) may be divided by KD and re-arranged as:
                    VRBGP        =                              VBGAP            KD                    =                                                                      1                  KD                                ⁢                VBE                ⁢                                                                  ⁢                1                            +                              (                                                      VBE                    ⁢                                                                                  ⁢                    1                                    -                                      VBE                    ⁢                                                                                  ⁢                    2                                                  )                                      =                                          VBE                ⁢                                                                  ⁢                1                ⁢                                  (                                      1                    +                                          1                      KD                                                        )                                            -                              VBE                ⁢                                                                  ⁢                2                                                                        (        3        )            The bandgap voltage VBGAP is then scaled down by the factor KD. This can provide a reversed bandgap voltage level VRBGP of about 200 mV.
Equation (3) may also be written as:
                    VRBGP        =                                            1              KD                        ⁢            VBE            ⁢                                                  ⁢            1                    +                      Vt            ⁢                                                  ⁢            ln            ⁢                                                  ⁢                          (                                                IC                  ⁢                                                                          ⁢                  1                  ×                  IS                  ⁢                                                                          ⁢                  2                                                  IS                  ⁢                                                                          ⁢                  1                  ×                  IC                  ⁢                                                                          ⁢                  2                                            )                                                          (        4        )            U.S. Pat. No. 7,411,443 discloses a high precision reversed bandgap voltage reference circuit. FIG. 1 illustrates an embodiment of this kind of reversed bandgap voltage reference circuit. First resistor R1 and second resistor R2 are coupled as a voltage divider which is connected between ground and first conductor 17. The circuit includes first transistor Q1 and second transistor Q2. The base of first transistor Q1 is coupled to the voltage divider to produce a first voltage VBE1(1+1/KD) between first conductor 17 and ground at node 16A. KD is the ratio of the resistances of first resistor R1 and second resistor R2. Second transistor Q2 is also coupled through respective resistors R4 and R5 between first conductor 17 and ground. The base of second transistor Q2 is coupled to first conductor 17. The circuit includes amplifier 12. The positive input of amplifier 12 is coupled to the collector of first transistor Q1 (node 16A) and the negative input of amplifier 12 is coupled to the collector of second transistor Q2 (node 16B). The output of amplifier 12 is coupled to a gate of MOSFET M1. MOSFET M1 is coupled between the supply voltage VDD and first conductor 17. Amplifier 12 controls conduction in MOSFET M1 so that the voltages on nodes 16A and 16B are equal. The reversed bandgap voltage VRBGP is output at the emitter of second transistor Q2. Due to the coupling of the transistors Q1 and Q2, the voltage at first conductor is VBE1(1+1/KD)=VBE2+VRBGP. This provides that VRBGP=VBE1(1−1/KD)−VBE2 as shown by Equation (3).
The circuits described in U.S. Pat. No. 7,411,443 provide exceptional accuracy and they are capable of operating with supply voltage levels below 1 V. However, these circuits are rather complex and they require bipolar transistors with a high gain of at least 40. The collectors of these transistors cannot be mandatorily tied to ground because of their substrate type as some technologies require. In particular, standard CMOS technologies provide only low gain of 3 to 4 and substrate type bipolar transistors.